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Mirrors > Home > ILE Home > Th. List > freq2 | GIF version |
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.) |
Ref | Expression |
---|---|
freq2 | ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frforeq2 4317 | . . 3 ⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐵𝑠)) | |
2 | 1 | albidv 1811 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠)) |
3 | df-frind 4304 | . 2 ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) | |
4 | df-frind 4304 | . 2 ⊢ (𝑅 Fr 𝐵 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠) | |
5 | 2, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1340 = wceq 1342 FrFor wfrfor 4299 Fr wfr 4300 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-in 3117 df-ss 3124 df-frfor 4303 df-frind 4304 |
This theorem is referenced by: weeq2 4329 |
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